Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency

• URL: http://arxiv.org/abs/2605.05942v1
• Date: Thu, 07 May 2026 09:52:45 GMT
• Title: Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency
• Abstract summary: We show that trainability varies substantially with architecture shape $(N,L)$ at fixed $E$.<n>We identify structural rank deficiency of the coefficient matching Jacobian $J$ as the mechanism responsible.
• Score: 3.1094222618006664
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging $N$ qubits with $L$ encoding layers each -- sharing encoding budget $E = NL$ -- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape $(N,L)$ at fixed $E$. We identify structural rank deficiency of the coefficient matching Jacobian $J$ as the mechanism responsible. For serial single-qubit architectures, we prove $\mathrm{rank}(J) \leq 2L+1$ regardless of parameter count $P$, with $\dim(\ker J) \geq P-(2L+1)$ growing without bound -- a phenomenon we term \emph{structural gradient starvation}: a growing fraction of parameters become structurally decoupled from the loss as $P$ increases at fixed $L$. Parallel architectures avoid this via independent phase trajectories, ensuring $σ_{\min}(J^{(\mathrm{par})}) > 0$ generically for $P \leq 2E+1$, so no parameter lies in $\ker J$. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves $R^2 \geq 0.95$ with $1.6$--$2.2\times$ fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.

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